Handbook of Wireless Networks and Mobile Computing, Edited by Ivan Stojmenovic ´ Copyright © 2002 John Wiley & Sons, Inc ISBNs: 0-471-41902-8 (Paper); 0-471-22456-1 (Electronic) CHAPTER Channel Assignment and Graph Labeling JEANNETTE C M JANSSEN Department of Mathematics and Statistics, Dalhousie University, Halifax, N.S., Canada 5.1 INTRODUCTION Due to rapid growth in the use of wireless communication services and the corresponding scarcity and high cost of radio spectrum bandwidth, it has become increasingly important for cellular network operators to maximize spectrum efficiency Such efficiency can be achieved by optimal frequency reuse, i.e., the simultaneous use of the same part of the radio spectrum by communication links in different locations of the network Optimal frequency reuse is constrained by noise levels, resulting from interference between communication links, that must be kept at acceptable levels (see [25]) In the previous chapter [28], the problem of assigning channels to minimize spectrum use while satisfying interference constraints was discussed in its simplest form In this form, each pair of cells in the network can either use the same channel simultaneously or not However, for networks based on frequency division (FDMA) or time division (TDMA), there can be a significant difference in the amount of interference between channels that are near each other in the radio spectrum and channels that are far apart This implies that the distance between cells that use channels close together in frequency must be greater than the distance between cells that use channels that are far apart The constraints for channel assignment resulting from this consideration are referred to as channel separation constraints As discussed in [28], a graph model can be used for the channel assignment problem The nodes of the graph correspond to cells or their base stations and the edges represent cell adjacency We assume that a fixed demand for channels is given for each cell, and that a channel assignment assigning exactly that many channels to the cell must be found The algorithms reviewed here apply to the static situation However, in many cases the same algorithms can also be used in the dynamic situation, where the demand for channels changes over time Algorithms based on a preassigned set of channels per node (such as Algorithms A and AЈ, described in Section 5.3) can be directly adapted to the dynamic case Other algorithms can be adapted if limited reassignment of the channels used to carry ongoing calls is permitted From another viewpoint, the static demand could represent the average or maximum possible demand for channels in the cell, and the fixed channel assignment based on this demand is expected to perform well even in the dynamic situa95 96 CHANNEL ASSIGNMENT AND GRAPH LABELING tion In the graph model given here, the demand is represented by a positive integer w(v), associated with each node v of the graph An assignment of integer values to the nodes of a graph so that certain conditions are satisfied is referred to as a graph labeling A coloring of a graph can thus be seen as a special case of a graph labeling, satisfying the condition that the labels of adjacent nodes must be distinct The framework of graph labeling gives us the possibility to incorporate the channel separation constraints We represent these constraints by a nonincreasing sequence of positive integer parameters c0, c1, , ck (so that c0 Ն c1 Ն Ն ck) Making the reasonable assumption that graph distance relates to physical distance between cells, we require that channels assigned to nodes (cells) at graph distance i from each other must have a separation of at least ci The constraint c0 represents the separation between channels assigned to the same cell and is referred to as the cosite constraint The constraints between different cells are referred to as intersite constraints Although the cosite constraint is often high compared to the other constraints, the intersite constraints most often take smaller values, especially one and two In somewhat confusing terms, an intersite constraint of one, which indicates that channels assigned to the corresponding cells must be distinct, is often referred to as a cochannel constraint An intersite constraint of two, which codifies the requirement that channels assigned to a pair of cells cannot be next to each other in the radio spectrum, is often called an adjacent-channel constraint Note further that graph labeling usually refers to an assignment of one channel per node, so that c0 is irrelevant In Section 5.3, we will show how graph labelings can be useful in finding algorithms for channel assignment problems with demands greater than We will now proceed with a formal definition of the model described, and a review of other relevant models 5.1.1 Graph Models The definitions and notations used in this chapter are consistent with those introduced in the previous chapter [28] For a general background on graph theory, the reader is referred to [8] A constrained graph G = (V, E, c0, , ck) is a graph G = (V, E) and positive integer parameters c0, , ck, c0 Ն c1 Ն Ն ck called its constraints The constraints represent the prescribed channel spacing for pairs of channels assigned to the same node or to different nodes More precisely, ci represents the constraint between nodes at graph distance i from each other The reuse distance of G equals k + 1, the minimum graph distance between nodes that can use the same channel For consistency, the constraint between nodes whose distance is at least the reuse distance is defined to be zero A constrained, weighted graph is a pair (G, w) where G is a constrained graph and w is a positive integral weight vector indexed by the nodes of G The component of w corresponding to node u is denoted by w(u) and called the weight of node u The weight of node u represents the number of calls to be serviced at node u We use wmax to denote max{w(v) | v ʦ V} and wmin to denote the corresponding minimum weight of any node in the graph For any set S ʕ V, we use w(S) to denote the sum of the weight of all nodes in S In the context of our graph model, a formal definition of a channel assignment can now be given A channel assignment for a constrained, weighted graph (G, w) where G = (V, E, 5.1 INTRODUCTION 97 c0, , ck) is an assignment f of sets of nonnegative integers (representing the channels) to the nodes of G satisfying the conditions: | f (u)| = w(u) for all u ʦ V i ʦ f (u) and j ʦ f (v) ⇒ |i – j| Ն cᐉ for all u, v ʦ V so that dG(u, v) = ᐉ The bandwidth used by a channel assignment is represented by its span The span S( f ) of a channel assignment f of a constrained weighted graph is the difference between the lowest and the highest channel assigned by f The span of a constrained, weighted graph (G, w) denoted by S(G, w), is the minimum span of any channel assignment for (G, w) The regular layouts often used for cellular networks can be modeled as subgraphs of an infinite lattice An n-dimensional lattice is a collection of points in ‫ޒ‬n by n that are linear integer combinations of the generating vectors e1, , en The graph corresponding to the lattice has the points as its nodes, and two nodes are adjacent precisely when one can be obtained from the other by adding a generating vector The linear layout of mobile networks for car phones running along highways can be modeled as a path that is a subgraph of the line lattice The line lattice is a one-dimensional lattice generated by e = (1) The line lattice is bipartite Another bipartite graph is the square lattice, which is generated by e2 = ( ) and e1 = ( ) A subgraph of the square lattice is called a bidimensional grid The type of graph most commonly used to model cellular networks is the hexagon graph Hexagon graphs are subgraphs of the triangular lattice, which is generated by e1 = 1/2 ( ) and d = ( 1/2 ͙3 ) Hexagon graphs model the adjacencies between hexagonal cells in a ෆ regular cellular layout resembling a honeycomb This layout is popular with network designers since hexagons resemble the circular area around a transmitter where its signals can be comfortably received In urban networks, hexagonal networks cannot always be achieved because of limitations of terrain On the other hand, networks based on satellite systems operate with almost perfect hexagon-based networks Channel assignment algorithms are often built on a basic assignment of one channel per node, known as a graph labeling Formally, a graph labeling of a graph G = (V, E) is any assignment f : V Ǟ ‫ ގ‬of integers to the nodes The labeling f satisfies the constraints c1, ck if for all pairs of nodes u, v at distance d = dG(u, v) Յ k from each other, |f(u) – f(v)| Ն cd The span S( f ) of a labeling f is defined as S( f ) = max f (V), the value of the highest label assigned by f Note: if the lowest label used is zero, then the definition of the span of a graph labeling is consistent with that of a channel assignment In order to use a graph labeling to find channel assignments for weight vectors with components greater than 1, one must know the “offset” that is needed when the labeling is repeated This notion is captured in the definition of cyclic span The cyclic span of a labeling f is the smallest number M such that, for all pairs of nodes u, v at graph distance d from each other (d Յ k), |f(u) – f(v)| Ն M – cd The first paper to consider graph labelings for constraints c1, ck, [15], referred to them as L(c1, ck)-labelings The specific case of a graph labeling with c1 = 2, c2 = is called a radio coloring, [10], or ␭-coloring (see for example [4]) Labelings for graphs with constraints c1, c2, , ck = k, k – 1, , 2, 1, where k is the diameter of the graph, are called radio labelings, and were studied by Chartrand et al [6] 98 CHANNEL ASSIGNMENT AND GRAPH LABELING A related model is based on a representation of the constraints by the minimum distance that must exist between pairs of cells that are assigned channels a fixed distance apart in the radio spectrum (see [26, 37], for example) More precisely, a set of nonincreasing parameters d0, d1, , dk is given and a channel assignment f has to fulfill the condition that for any pair of nodes (cells) u and v i ʦ f (u) and j ʦ f (v) and |i – j| = ᐉ ⇒ d(u, v) > dᐉ The distance d(u, v) can either be used to mean the physical distance between the corresponding base stations or the graph distance between the nodes If graph distance is used, then the correspondence between this model and our model is that dᐉ = min{i|ci < ᐉ} for ᐉ > 0, d0 = k + 1, and ci = min{ᐉ|i > dᐉ} In this model, d0 equals the reuse distance Another model assumes that for each pair of adjacent nodes u, v a separation constraint cu,v is given (see for example [30]) A channel assignment f must satisfy the condition that, for each pair u, v i ʦ f (u) and j ʦ f (v) ⇒ |i – j| Ն cu,v This model is useful if geographical distance is not the only cause of interference, a case often seen in urban environments where additional factors like obstructing structures and antenna placement affect interference levels In such cases, the interference information is often obtained from measurements, and is reported in the form of an interference matrix with entries for each pair u, v The above model is more general than the one used in this chapter However, the latter model is consistent with the one described above This is easily seen by setting cu,v = ci for all pairs of nodes u and v at graph distance i Յ k from each other, and cu,v = for all other pairs of nodes Most of the lower bounding techniques described in Section 5.2 originally referred to this general model 5.1.2 Algorithmic Issues In this chapter, only channel assignment algorithms for which theoretical bounds on their performance have been established are discussed The papers not considered here roughly fall into three categories The first of these propose heuristics and give experimental results without theoretical analysis The second group focuses on implementation issues arising from specific technologies and protocols, and the final group gives exact solutions to certain specific instances by using combinatorial optimization methods such as integer programming The term “performance ratio” refers here to the asymptotic performance ratio Hence, a channel assignment f is said to be optimal for a weighted constrained graph G if S( f ) = S(G, w) + O(1) The span is assumed to be a function of the weights and the size of the graph, so the O(1) term can include terms dependent on the constraints c0, c1, , ck An approximation algorithm for channel assignment has performance ratio k when the span of the assignment produced by the algorithm on (G, w) is at most kS(G, w)+ O(1) The version of the channel assignment problem considered here is a generalization of 5.2 LOWER BOUNDS 99 the graph coloring problem, which is well known to be NP-complete for general graphs A reduction to Hamiltonian paths shows that channel assignment is NP-complete even for graphs of diameter with constraints c1 = 2, c2 = This was proved in the seminal paper on graph labelings by Griggs and Yeh [15] (The same result, with the same proof, was presented without reference to the original result six years later by Fotakis and Spirakis in [11].) McDiarmid and Reed [27] have proved that multicoloring is NP-hard for hexagon graphs, which implies that channel assignment for hexagon graphs with general constraints is NP-hard The proof involves a reduction of the multicoloring problem to the problem of coloring a planar graph The proof can easily be adapted to demonstrate the NP-hardness of channel assignment for hexagon graphs under any specific choice of constraints c0, c1, , ck The algorithms described in this chapter are all static This means that such algorithms attempt to find the best possible channel assignment for one particular constrained graph and one particular weight vector In realistic networks, the demand for calls changes continuously However, as indicated in the previous chapter, there is a strong connection between on-line algorithms, which can account for changes in weights, and static algorithms The algorithms presented assume a global control mechanism that implements the assignment in the whole graph In reality, it may be desirable to implement channel assignment in a distributed manner, i.e., the decision on the assignment of channels can be taken at each node independently or after limited consultation between the node and its local neighborhood Once again, little of the research presented here targets the distributed case specifically However, I have indicated for each algorithm what information must be present at a node and how much communication between nodes is needed It can therefore be quickly determined which algorithms can be implemented so that each node finds its own channel assignment 5.2 LOWER BOUNDS In order to evaluate any algorithm and to be able to give bounds on its performance ratio, it is essential to have good lower bounds Some lower bounds, such as those based on the maximum demand in a cell, are straightforward to obtain Others can be derived from representations of the channel assignment problem as a graph coloring problem or a traveling salesman problem In this section, I will give an overview of the lower bounds available for channel assignment with constraints An early paper by Gamst [12] presents a number of lower bounds based on sets of nodes that have a prescribed minimum constraint between them More precisely, a dclique in a constrained graph G = (V, E, c0, c1, ) is a set of nodes so that for any pair of nodes u, v, dG(u, v) Յ d Note that a d-clique corresponds to a clique in Gd, the graph obtained from G by adding edges between all pairs of nodes with distance at most d in G Any two nodes in a d-clique have constraint at least cd between them, and thus any two channels assigned to nodes in the d-clique have to have separation at least cd This leads 100 CHANNEL ASSIGNMENT AND GRAPH LABELING directly to the following bound, adapted from [12] For any constrained, weighted graph (G, w), where G = (V, E, c0, c1, , ck) S(G, w) Ն max{cd w(C) – cd|C a d-clique of G} (5.1) For the special case d = 0, the clique consists of only one node and this bound transforms into a bound derived from the maximum weight on any node: S(G, w) Ն max{c0 w(v) – c0|v ʦ V} (5.2) The clique bound can be extended to a bound based on the total weight of a graph and the size of a maximum independent set A d-independent set in a constrained graph G is an independent set in Gd In other words, it is a set of nodes so that for any pair of nodes u, v, dG(u, v) > d If ␣d(G) denotes the maximum size of a d-independent set in G, then in any channel assignment for G, at most ␣d nodes can obtain channels from any interval {k, k + 1, , k + cd – 1} This leads to the following bound, stated slightly differently in [33]: S(G, w) Ն max{cd w(H)/␣d(H) – cd|H a subgraph of G} (5.3) 5.2.1 Traveling Salesman Bounds Several authors ([22, 15, 17, 31, 33]) have noted that the channel assignment problem can be reframed as a generalization of the traveling salesman problem (TSP) For any channel assignment of a graph with weight one on every node, an enumeration of the nodes in nondecreasing order of the assigned channels will constitute an open TSP tour (Hamiltonian path) of the nodes The difference between the channels assigned to two consecutive nodes in the tour is at least equal to the constraint between the nodes Hence, the span of the assignment is at least equal to the cost of the tour, with the cost of traveling between two nodes u and v being the constraint between these two nodes Therefore, the cost of an optimal TSP tour is a lower bound on the span of the channel assignment If the weights are greater than one, one can derive a similar bound from a generalized TSP problem Here, the goal is to find a minimum cost tour such that every node v is visited w(v) times Note that this corresponds to a regular TSP if every node v is expanded into a clique of size w(v), where the cost between any two nodes in this clique is defined to be c0, whereas the nodes in cliques corresponding to different nodes of the original graph inherit the cost between those original nodes If the constraints in a graph have the property that ci + cj Ն ck, for all i, j, k so that i + j Ն k, then the corresponding TSP problem is Euclidean, and the cost of the optimal tour equals the cost of the best channel assignment Note that this property holds for nonincreasing constraints c0, , ck precisely if 2ck Ն c0 For any constrained, weighted graph (G, w) (where G = (V, E, c0, , ck)), let cG ʦ ‫ޚ‬V×V be the vector that represents the constraints between pairs of nodes of G Given a set of nodes V, a weight vector w ʦ ‫ޚ‬V and a cost vector c ʦ ‫ ޚ‬V×V, let TSP(V, w, c) be the + 5.2 LOWER BOUNDS 101 cost of the minimum traveling salesman tour through V, where each node v is visited w(v) times, and costs are given by c Then the following bound, first given in [33], holds: S(G, w) Ն max{TSP(U, w, cG) – c0|U ʕ VG} (5.4) (Vectors w and cG are considered to be restricted to U.) The minimal TSP tour can be as hard to compute as the optimal channel assignment, so this bound is only of practical interest for relatively small channel assignment problems However, the TSP approach can be used to find a lower bound that is easy to calculate As mentioned in [33], the lower bound for the TSP given by Christofides (see for example [7]), which is derived from minimum spanning trees and is easy to compute, may be used to approximate the TSP bound A linear programming relaxation of the generalized TSP problem can also be used to derive lower bounds for channel assignment (see [20]) A TSP tour is seen as a collection of edges, so that each node is covered by exactly two of these edges Not every such edge cover corresponds to a TSP tour, but the minimum edge cover will constitute a lower bound for the TSP tour Moreover, a fractional relaxation of the edge cover problem will give lower bounds that are easy to compute Given a set of nodes V, a weight vector w ʦ ‫ޚ‬V and a cost vector c ʦ ‫ ޚ‬V×V, a fraction+ al edge cover is a vector y ʦ ‫ޑ‬V×V so that ⌺w yvw Ն for each vʦ V The cost of a fractional edge cover y is defined as ⌺vwʦE c(vw)yvw Letting EC*(V, w, c) be the minimum cost of any fractional edge cover of node set V, with weight and cost vectors w and c, respectively The following is a relaxation of the TSP bound: S(G, w) Ն EC*(V, w, cG) – c0 (5.5) This bound can be refined by adding some of the subtour constraints, which explicitly forbid solutions that consist of disconnected cycles Potentially, there are an exponential number of subtour constraints, but in practice a small number of subtour constraints, added in an iterative manner, will lead to good approximations of the value of the TSP tour The bound obtained in this way is referred to as the Held–Karp bound In [23] it is shown that for a wide variety of randomly generated instances, the cost of the optimal tour is on average less than 0.8% of the Held–Karp bound, and for real-world instances the gap is almost always less than 2% A version of this approach to the TSP bound was implemented by Allen et al.; computational results are presented in [1] The edge cover problem can also be analyzed using polyhedral methods, to yield a family of explicit lower bounds (see [16]) One specific edge cover bound was used in [19] to solve the “Philadelphia problem,” a benchmark problem from the early days of the frequency assignment problem 5.2.2 Tile Cover Bounds Bounds derived from the TSP and its relaxations may not be very good if ci + cj < ck for some indices i, j, k such that i + j Ն k In this case, a piece of the tour consisting of three consecutive nodes u, v, w so that d(u, v) = i, d(v, w) = j, and d(u, w) = k will con- 102 CHANNEL ASSIGNMENT AND GRAPH LABELING tribute an amount of ci + cj to the tour, whereas the separation between channels at u and w must be at least ck In this case, one approach is to break a channel assignment into chunks of nodes that receive consecutive channels Such chunks will be referred to as “tiles,” and the cost of a tile will be related to the minimum bandwidth required to assign channels to its nodes The channel assignment problem is thus reduced to a problem of covering the nodes with tiles, so that each node v is contained in at least w(v) tiles The fractional version of the tile cover problem can be easily stated and solved, and then used to bound the minimum span of a channel assignment Since the tile cover method is not widely known, but gives promising results, we shall describe it in some detail in this section The tile cover approach was first described in [20] The method can be outlined as follows For a constrained graph G, a set T of possible tiles that may be used in a tile cover is defined All tiles are defined as vectors indexed by the nodes of G A collection of tiles (multiple copies allowed) can be represented by a nonnegative integer vector y ʦ ‫ ޚ‬T, where y(t) represents the number of copies of tile t present in the + tiling A tile cover of a weighted constrained graph (G, w) is such a vector y with the property that ⌺tʦT y(t)t(v) Ն w(v) for each node v of G A cost c(t) is associated with each tile t ʦ T The cost of each tile t is derived from the minimal span of a channel assignment for (G, t) plus a “link-up” cost of connecting the assignment to a following tile This “link-up” cost is calculated using the assumption that the same assignment will be repeated The cost of a tile cover y is defined as c(y) = ⌺tʦT y(t)c(t) The minimal cost of a tile cover of a weighted, constrained graph (G, w) will be denoted by ␶(G, w) In order to derive lower bounds from tile covers, it must be established that for the graphs and constraints under consideration S(G, w) Ն ␶ (G, w) – k where k is a constant that does not depend on w The problem of finding a minimum cost tile cover of (G, w) can be formulated as an integer program (IP) of the following form: Minimize ⌺tʦT c(t)y(t) subject to: ⌺tʦT t(v)y(t) Ն w(v) y(t) Ն y integer (v ʦ V) (t ʦ T ) The linear programming (LP) relaxation of this IP is obtained by removing the requirement that y must be integral Any feasible solution to the resulting linear program is called a fractional tile cover The minimum cost of a fractional tile cover gives a lower bound on the minimum cost of a tile cover By linear programming duality, the maximum cost of the dual of the above LP is equal to the minimum cost of a fractional tile cover Thus, any vector that satisfies the inequalities of the dual program gives a lower bound on the cost of a minimum fractional tile cover, and therefore on the span of the corresponding constrained, weighted graph The maxi- 5.2 LOWER BOUNDS 103 mum is achieved by one of the vertices of the polytope TC(G) representing the feasible dual solutions and defined as follows: Ά TC(G) = x ʦ ‫ޑ‬V : Α t(v)x(v) Յ c(t) for all t ʦ T + vʦV · A classification of the vertices of this polytope will therefore lead to a comprehensive set of lower bounds that can be obtained from fractional tile covers For any specific constrained graph, such a classification can be obtained by using vertex enumeration software, e.g., the package lrs, developed by Avis [2] In [18], 1-cliques in graphs with constraints c0, c1 were considered In this case the channel assignment was found to be equivalent to the tile cover problem Moreover, the fractional tile cover problem is equivalent to the integral tile cover problem for 1-cliques, leading to a family of lower bounds that can always be attained None of the bounds was new Two bounds were clique bounds of the type mentioned earlier The third bound was first given by Gamst in [12], and can be stated as follows: S(G, w) Ն max{c0w(v) + (␮c1 – c0)w(C – v) – c0|C a clique of G, v ʦ C} (5.6) where ␮ is such that (␮ – 1)c1 < c0 Յ ␮c1 The tile cover approach led to a number of new bounds for graphs with constraints c0, c1, c2 The bounds are derived from so-called nested cliques A nested clique is a d1-clique that contains a d2-clique as a subset (d2 < d1) It is characterized by a node partition (Q, R), where Q is the d2-clique and R contains all remaining nodes A triple (k, u, a) will denote the constraints k = c0, u = cd2, and a = cd1 in a nested clique Note that in a nested clique with node partition (Q, R) with constraints (k, u, a), every pair of nodes from Q has a constraint of at least u, while the constraint between any pair of nodes in the nested clique is at least a The following is a lower bound for a nested clique (Q, R) with parameters (k, a, u): S(G, w) Ն aΑ w(v) + uΑ w(v) – u vʦQ (5.7) vʦR This bound was first derived in [12] using ad-hoc methods The same bound can also be derived using edge covers Using tile covers, a number of new bounds for nested cliques with parameters (k, u, 1) are obtained in [22] The following is a generalization of bound (5.6) (The notation wQmax and wRmax is used to denote the maximum weight of any node in Q and R, respectively.) S(G, w) Ն (k – ␮␦)wQmax + ␦ Α w(v) + ⑀ Α w(v) – k vʦQ where k ␮ = ᎏ , ␦ = (␮ + 1)u – k u   vʦR (5.8) 104 CHANNEL ASSIGNMENT AND GRAPH LABELING and ␦ 2u + ␮␦ – ␮ ⑀ = ᎏᎏ , ᎏᎏ k – 2u + k+1 Ά · Bound (1.3), obtained from the total weight on a clique, was extended, leading to ΂ Α w(v) + w ΃ + ᎏ k–1 k–u S(G, w) Ն u Α Rmax vʦQ w(v) – k (5.9) vʦR,v vRmax A bound of (2u – 1)wQmax + ⌺vʦRw(v) – ␬ for nested cliques where Q consists of one node was obtained in [34] This bound is generalized in [22] to all nested cliques: S(G, w) Ն (2u – 1)wQmax + ␯ Α vʦQ,v vQmax w(v) + Α w(v) – k (5.10) vʦR where u–1 ␦–1 ␯ = u – max ᎏ , ᎏ ␮ ␮–1 Ά · Finally, we mention the following two tile cover bounds from [22] for nested cliques with parameters (k, u, a): S(G, w)) Ն (3u – k + 2␦) Α w(v) + (k – 2␦)wRmax + ␦ Α w(v) – k vʦQ (5.11) vʦR where ␦ = 3a – k, and ΂ Α w(v) + w ΃ + ᎏ 3a – u S(G, w) Ն u Rmax vʦV Α w(v) – k (5.12) vʦR,v vRmax In [40], a bounding technique based on network flow is described Since no explicit formulas are given, it is hard to compare these bounds with the ones given in this section However, in an example the authors of [40] obtain an explicit lower bound that can be improved upon using edge covers [1] or tile cover bounds [22] 5.3 ALGORITHMS In this section, an overview is given of algorithms for channel assignment with general constraints Some of these algorithms are adaptations of graph multicoloring algorithms as described in the previous chapter and others are based on graph labeling An overview of the best-known performance ratios of algorithms for different types of graphs and constraints is presented in Table 5.1 5.3 ALGORITHMS 105 TABLE 5.1 An overview of the performance ratios of the best known algorithms for different types of graphs A * indicates that the performance ratio depends heavily on the constraints; see the text of Section 5.3 for details Constraints Bipartite graphs c0, c1 : c0 Ն 2c1 c0, c1 : c0 > 2c1 Paths c0, c1, c2 c0, c1, c2, c3 Performance ratio 1 Reference [14] max{1, (2c1 + c2)/c0 * [42, 13] [39] Bidimensional grid c0, c1, c2 c0, c1, c2, c3 max{1, (2c1 + 3c2)/c0} max{1, 5c1/c0, 10c2/c0} [13, 39] [3] Odd cycles (length n) c0, c1 : c0 Ն (2nc1)/(n – 1) c0, c1 : 2c1 Յ c0 < (2nc1)/(n – 1) c0, c1 : c1 Յ c0 < 2c1 c0, c1, c2 1 + 1/(4n – 3) + 1/(n – 1) max{1, 3c1/c0, 6c2/c0} [21] [21] [21] [15] max{1, 3c1/c0} p(v), then v receives the additional channels f (v)c1 + 2p(v)c1 + ic0, i = 0, , w(v) – p(v) – The span of the assignment above is at most max(uv)ʦE {c0w(u) + (2c1 – c0)w(v)} It follows from lower bound 5.6 that the algorithm is (asymptotically) optimal In fact, [14] gives a more detailed version of the algorithm above that is optimal in the absolute sense For higher constraints, the only results available are for graph labelings of specific bipartite graphs Van den Heuvel et al [39] give labelings by arithmetic progression for subgraphs as the line lattice (paths) Such labelings only have n (the cyclic span) and a1 = a as parameters If f is such a labeling, then a node v defined by the vector me will have value f (v) = ma mod n The parameters of the labelings are displayed in the table below These labelings are optimal in almost all cases The exception is the case where there are three – constraints c1, c2, and c3, and 2c2 – c3 Յ c1 Յ (1 )c2 + c3 For this case, a periodic labeling not based on arithmetic progressions is given in the same paper Constraints c1, c2 c1, c2, c3 : c1 Ն c2 + c3 c1, c2, c3 : c2 + (1/3)c3 Յ c1 Յ c2 + c3 c1, c2, c3 : c1 Յ c2 + (1/3)c3 n a 2c1 + c2 2c1 + c2 3c2 + 2c3 3c1 + c3 c1 c1 c2 + c3 c1 For paths of size at least five, these labelings include the optimal graph labeling satisfying constraints c1 = 2, c2 = given by Yeh in [42], and the path labelings for general constraints c1, c2 by Georges and Mauro in [13] Note that Algorithm AЈ, used with any of these labelings with cyclic span n, has a performance ratio of max{1, n/c0} The near-optimal labeling for unit interval graphs given in [32] can be applied to paths with constraints c1, c2, , c2r, where c1 = c2 = = cr = and cr+1 = c2r = 1, to give a labeling with cyclic span 2r + Using this labeling in Algorithm AЈ leads to a performance ratio of max{1, (2r + 1)/c0} Van de Heuvel et al [39] also give an optimal labeling by arithmetic progression for the square lattice and constraints c1, c2 The labeling given has cyclic span n = 2c1 + 3c2 and is defined by the parameters a1 = c1, a2 = c1 + c2 The square lattice is the Cartesian product graph of two infinite paths, and similar labelings can also be derived from the results on products of paths given in [13] Bertossi et al [3] give a labeling for constraints c1 = 2, c1 = c3 = of span and cyclic span 10 This labeling can be transformed into a labeling for general c1, c2, c3 as follows Let c = max{c1/2, c2}, and let f be the labeling for c1, c2, c3 = 2, 1, Let f Ј(u) = cf (u) It is easy to check that f Ј is a labeling for c1, c2, c3 of cyclic span 10c Using this labeling with Algorithm AЈ gives a performance ratio of max{1, 5c1/c0, 10c2/c0} The same authors give a labeling for bidimensional grids with constraints c1 = 2, c2 = 1, which is just a special case the labeling by arithmetic progression given above The same authors also give labelings for graphs they call hexagonal grids, with constraints c1, c2 = 2, and c2, c1, c3 = 2, 1, Hexagonal grids are not to be confused with hexagon graphs, which will be discussed in Section 5.3.3 In fact, hexagonal grids are 108 CHANNEL ASSIGNMENT AND GRAPH LABELING subgraphs of the planar dual of the infinite triangular lattice Hexagonal grids form a regular arrangement of cycles, and are bipartite Labelings for the hypercube Qn were described and analyzed in [15, 24, 41] Graph labelings for trees with constraints c1, c2 = 2, were treated in [5] and [15] These labelings are obtained using a greedy approach, which is described in Section 5.3.4 5.3.2 Odd Cycles Channel assignment on odd cycles was first studied by Griggs and Yeh in [15] The authors give a graph labeling for constraints c1, c2 = 2, of span and cyclic span The labeling repeats the channels 0, 2, along the cycle, with a small adaptation near the end if the length of the cycle is not divisible by As described in the previous section, this labeling can be used for general constraints c1, c2 if all values assigned by the labeling are multiplied by max{c2, c1/2} Using Algorithm AЈ, this leads to an algorithm with performance ratio max{1, 3c1/c0, 6c2/c0} In [21], three basic algorithms for odd cycles are combined in different ways to give optimal or near-optimal algorithms for all possible choices of two constraints c0 and c1 The first of the three algorithms in [21] is based on a graph labeling that satisfies one constraint c1 This labeling has cyclic span cR = 2nc1/(n – 1) It starts by assigning zero to the first node, and then adding c1 (modulo cR) to the previously assigned channel and assigning this to the next node in the cycle At a certain point, this switches to an alternating assignment This labeling is then used repeatedly, as in Algorithm AЈ Since this particular form of Algorithm AЈ will be used to describe the further results in this chapter, I will state it explicitly below Algorithm C (for odd cycles) Let G = (V, E, c0, c1) be a constrained cycle of n nodes, where n > is odd, and w be an arbitrary weight vector Fix s = max{c0, cR} Let the nodes of the cycle be numbered {1, , n}, numbered in cyclic order, where node is a node of maximum weight in the cycle Let m > be the smallest odd integer such that s Ն 2m/(m – 1)c1 (it can be shown that such an integer must exist) ASSIGNMENT: To each node i, the algorithm assigns the channels b(i) + js, where j = 0, , w(i) – 1, and the graph labeling b : V Ǟ [0, s – 1] is defined as follows: Ά (i – 1)c1 mod s b(i) = (m – 1)c1 mod s when Յ i Յ m, when i > m and i is even, when i > m and i is odd Note that this algorithm can only be implemented in a centralized way, since every node must know all weights, in order to calculate m, and so determine its initial assignment value The second algorithm is a straightforward adaptation of the optimal algorithm for multicoloring an odd cycle, described in [29] and discussed in the previous chapter The span used by this algorithm is ␻/2s 5.3 ALGORITHMS 109 Algorithm D (for odd cycles) Let G = (V, E, c0, c1) be a constrained cycle of n nodes, where n > is odd, and w be an arbitrary weight vector Fix s = max{c0, 2c1}, and ␻ = max{2⌺vʦVw(v)/(n – 1), 2wmax} Let f be an optimal multicoloring of (G, w) using the colors {0, 1, , ␻ – 1} Such an f exists since ␹(G, w) Յ ␻ ASSIGNMENT: For each node v, replace each color i in f (v) with the channel fi, where fi = Άis + (i – ␻)s c – if i Յ ␻\2 – 1, otherwise Algorithms C and D only give good assignments for weight vectors with specific properties, but they can be combined to give near-optimal algorithms for any weight vector How they are combined will depend on the relation between the parameters First, note that Algorithm C is optimal if c0 Ն cR = 2nc1/(n – 1) If 2c1 Յ c0 < cR, then Algorithms A, C, and D can be combined to give a linear time algorithm with performance ratio + 1/(4n – 3), where n is the number of nodes in the cycle The algorithm is described below Given a weight vector w, compute ␦ = ⌺vʦV w(v) – (n – 1)wmax If ␦ Յ 0, Algorithm D is used, with spectrum [0, c0 wmax] The span is at most c0wmax, which is within a constant of lower bound (5.2), so the assignment is optimal If instead ␦ > 0, Algorithm C is combined with either Algorithm A or D to derive an assignment Denote by f1 the assignment computed by Algorithm C for (G, wЈ) where wЈ(v) = min{w(v), ␦} This assignment has span at most cR␦ Consider the remaining weight w after this assignment Clearly wmax = wmax – ␦ We ෆ ෆ will denote by f2 the assignment for (G, w), and compute it in two different ways dependෆ ing on a key property of ෆ If there is a node v with w(v) = at this stage, we have a biparw ෆ tite graph left Then f2 is the assignment computed by Algorithm A for (G, ෆ) This assignw ment has a span of at most c0wmax ෆ If all nodes have nonzero weight, then Algorithm D is used to compute f2, the assignment for (G, w) It can be shown that in this case, ␻ = 2wmax, so this assignment also has a ෆ ෆ span of at most c0␻/2 = c0wmax Thus, in either case, f2 has span at most c0wmax ෆ The two assignments f1 and f2 are then combined by adding cR␦ + c0 to every channel in f2, and then merging the channel sets assigned by f1 and f2 at each node This gives a final assignment of span at most (cR – c0) ␦ + c0wmax + c0 Using the lower bounds (5.2) and (5.3), it can be shown that the performance ratio of the algorithm is as claimed If c0 < 2c1, Algorithms B and C can be combined into a linear time approximation algorithm with performance ratio + 1/(n – 1), where n is the number of nodes in the cycle The combination algorithm is formed as follows First, find the assignment f1 computed by Algorithm C for (G, wЈ) where wЈ(v) = wmin for every node v Then, find the assignment f2 computed by Algorithm B for (G, wЈЈ) where wЈЈ(v) = w(v) – wmin Finally, combine the two assignments by adding cRwmin + c0 to each channel of f2 and then merging the channel sets assigned by f1 and f2 Using bound 1.6, it can be shown that the algorithm has performance ratio + 1/(n – 1) as claimed 110 CHANNEL ASSIGNMENT AND GRAPH LABELING In [13], optimal graph labelings for odd cycles with constraints c1, c2 are given If c1 > 2c2, or c1 Յ 2c2 and n ϵ mod 3, the span is 2c1, and the cyclic span is 3c1 Using Algorithm AЈ in combination with this labeling gives a performance ratio of max{1, 3c1/c0} For the remaining case, the span is c1 + 2c2 and the cyclic span is c1 + 3c2, leading to a performance ratio for Algorithm AЈ of max{1, (c1 + 3c2)/c0} In [3], Bertossi et al give a graph labeling for cycles of length at least with constraints c1, c2, c3 = 2, 1, The span of the labeling is 4, and its cyclic span is Adapting this labeling to general parameters c1, c2, c3 and using Algorithm AЈ gives a performance ratio of max{1, 3c1/c0, 6c2/c0} 5.3.3 Hexagon Graphs The first labelings for hexagon graphs were labelings by arithmetic progression given by van den Heuvel et al in [39] The labelings, as defined by their parameters a1, a2, and n, are given in the table below Parameters c1 Ն 2c2 (3/2)c2 Յ c1 Յ 2c1 c1 Յ (3/2) c2 n a1 a2 3c1 + 3c2 9c2 4c1 + 3c2 2c1 + c2 5c2 2c1 + c2 c1 2c2 c1 It can be easily seen that hexagon graphs admit a regular coloring with three colors Hence Algorithm A will be optimal for constraints c0, c1 so that c0 Ն 3c1 A channel assignment algorithm for hexagon graphs with constraints c0, c1 = 2, with performance ratio 4/3 was given in [36] In [21], further approximation algorithms for hexagon graphs and all values of constraints c0, c1 are given All algorithms have performance ratio not much more than 4/3, which is the performance ratio of the best known multicoloring algorithm for hexagon graphs (see [28]) The results are obtained by combining a number of basic algorithms for hexagon graphs and bipartite graphs The algorithm described below is similar to the one in [36] Algorithm E (for 3-colorable graphs) Let G = (V, E, c0, c1) be a constrained graph, and w be an arbitrary weight vector Fix s = max{c1, c0/2} and T Ն 3wmax, T a multiple of Let f : V Ǟ {0, 1, 2} be a base coloring of G Denote base colors 0, 1, as red, blue and green, respectively A set of red channels is given, consisting of a first set R1 = [0, 2s, , (T/3 – 2)s] and a second set R2 = [(T/3 + 1)s + c0, (T/3 + 3)s + c0, , (2T/3 – 1)s + c0] Blue channels consist of first set B1 = [(T/3)s + c0, (T/3 + 2)s + c0, , (2T/3 – 2)s + c0] and second set B2 = [(2T/3 + 1)s + 2c0, (2T/3 + 3)s + 2c0, , (T – 1)s + 2c0], and green channels consist of first set G1 = [(2T/3)s + 2c0, (2T/3 + 2)s + 2c0, , (T – 2)s + 2c0] and second set G2 = [s, 3s, , (T/3 – 1)s] ASSIGNMENT: Each node v is assigned w(v) channels from those of its color class, where the first set is exhausted before starting on the second set, and lowest numbered channels are always used first within each set 5.3 ALGORITHMS 111 Note that the spectrum is divided into three parts, each containing T/3 channels, with a separation of s between consecutive channels The first part of the spectrum consists of alternating channels from R1 and G2, the second part has alternating channels from B1 and R2, and the third part has alternating channels from G1 and B2 The span used by Algorithm E equals sT + 2c0 = max{c1, c0/2}T + 2c0, where T is at least 3wmax To obtain the optimal algorithms for hexagon graphs and different values of the parameters c0, c1, Algorithm E is modified and combined with Algorithms A and B Algorithm A for hexagon graphs has a performance ratio of max{1, 3c1/c0} As noted, when c0 Ն 3c1 the algorithm is optimal When c0 Ն (9/4)c1, the performance ratio of equals 3c1/c0, which is at most 4/3 For the case where 2c1 < c0 Յ (9/4)c1, a combination of Algorithms A for hexagon graphs and Algorithm E followed by a borrowing phase and an application of Algorithm B results in an algorithm with performance ratio less than 4/3 + 1/100 The algorithm is outlined below Let D represent the maximum weight of any maximal clique (edge or triangle) in the graph It follows from lower bound (1.3) that S(G, w) Ն c1D – c1 For ease of explanation, we assume that D is a multiple of Phase 1: If D > 2wmax, use Algorithm A for hexagon graphs on (G, wЈ) where wЈ(v) = min{w(v), D – 2wmax} If D Յ 2wmax, skip this phase, and take wЈ(v) = for all v The span needed for this phase is no more than max{0, D – 2wmax}3c1 Phase 2: Let T = min{2wmax, 6wmax – 2D} Use Algorithm E on (G, wЈЈ), where wЈЈ(v) = min{w(v) – wЈ(v), T/3}, taking T as defined The span of the assignment is min{2wmax, (6wmax – 2D)}c0/2 + 2c0 It follows from the description that after this phase, in every triangle there is at least one node that has received a number of channels equal to its demand Phase 3: Any node that has still has unfulfilled demand tries to borrow channels assigned in Phase from its neighbors according to the following rule: red nodes borrow only from blue neighbors, blue from green, and green from red A red node v with w(v) > wЈ(v) + wЈЈ(v), where wB(v) is the maximum number of channels used during Phase by any blue neighbor of v, receives an additional min{w(v) – wЈ(v) – wЈЈ(v), T/3 – wB(v), T/6} channels from the second blue channel set B2, starting from the highest channels in the set A similar strategy is followed for blue and green nodes It can be shown that the graph induced by the nodes that still have unfulfilled demand after this phase is bipartite Phase 4: Let w denote the weight left on the nodes after the assignments of the first three ෆ phases Use Algorithm A to find an assignment for (G, w), which has a span of c0wmax ෆ ෆ The assignments of all four phases are then combined without conflicts, as in the theorems for odd cycles The final assignment has span at most (2wmax)c0/2 + c0(wmax/3) + ⌰(1) = (4/3)c0wmax + ⌰(1) It then follows from lower bounds (5.2) and (5.3) that the performance ratio equals + 3(c0 – 2c1)/c0 + (9c1 – 4c0)/3c1 When 2c1 < c0 Յ (9/4)c1, this is always less than 4/3 + 1/100 In particular, the maximum value is reached when c0/c1 = 3/͙ෆ When c0 = 2c1 or c0 = 9c1/4, the performance ratio is exactly 4/3 When c0 Յ 2c1, a linear time approximation algorithm with performance ratio 4/3 is obtained from an initial assignment by Algorithm E, followed by a borrowing phase and a 112 CHANNEL ASSIGNMENT AND GRAPH LABELING phase where assigned channels are rearranged in the spectrum, and finally an application of Algorithm B The algorithm follows Let L = max{c0w(u) + (2c1 – c0)(w(v) + w(r))|{u, v, r} a triangle} and let T be the smallest multiple of larger than max{L, Dc1}/c1 It follows from lower bounds (5.6) and (5.3) that Tc1 – ⌰(1) is a lower bound for the span of any assignment Phase 1: Use Algorithm E on (G, wЈ) where wЈ(v) = min{w(v), T/3} and T is defined above In this case s, the separation between channels, equals c1, so the span of the assignment is Tc1 Phase 2: Any red node v of weight greater than T/3 borrows min{w(v) – T/3, T/3 – wB(v), T/6} channels, where wB(v) is the maximum weight of any blue neighbor of v The channels are taken only from the second blue channel set, and start with the highest channels Blue and green nodes borrow analogously, following the borrowing rules given earlier (red Ǟ blue Ǟ green Ǟ red) Phase 3: Any red node v of weight more than T/3, whose blue neighbors have weight at most T/6, will squeeze their assigned channels from their second set as much as possible More precisely, the last T/6 – wB(v) channels assigned to v from R2 are replaced by min{w(v) – T/3 – wB(v), 2c1/c0(T/6 – wB(v))} channels with separation c0 which fill the part of the spectrum occupied by the last T/6 – wB(v) channels of R2 For example, let T = 24, c0 = 3, and c1 = Suppose v is a red corner node with at least two green neighbors, where w(v) = 13 and let wB(v) = In Phase 1, v received the channels 21, 25, 29, 33 from the set R2, whereas at least one blue neighbor of v received the channel 19 from B1 and no other channels from B1 or B2 were used by any neighbor of v Then in Phase 2, v borrows all four blue channels in B2, and in Phase 3, squeezes the part of the spectrum [21, 33] of R2 to get five channels In particular, it uses the channels 21, 24, 27, 30, 33 instead of the four channels mentioned above The reader can verify that in this example, cosite and intersite constraints are respected Phase 4: Let ෆ be the weight vector remaining after Phase It can be shown that the w graph induced by the nodes with positive remaining weight is bipartite We use Algorithm B to find an assignment for (G, ෆ), which has a span of LЈ = max{c0w(u) + (2c1 – w ෆ c0)w(v)|(u, v) ʦ E} ෆ The assignments of different phases are then combined without causing conflicts, in the same way as described before, to give a final assignment of span at most (4/3) Tc1 + ⌰(1) From the definition of T, we have that Tc1 – ⌰(1) is a lower bound, which gives the required performance ratio of 4/3 In [3], a labeling is given for hexagon graphs with constraints c1, c2, c3 = 2, 1, (Hexagon graphs are referred to as cellular grids in this paper.) The labeling has a span of 8, which is proven to be optimal, and a cyclic span of Moreover, when examined it can be determined that this labeling is, in fact, a labeling by arithmetic progression, with parameters n = 9, a = 2, b = It therefore follows from the results of van de Heuvel et al that the labeling is optimal, since is the optimal span even for constraints c1, c2 = 2, This la- 5.3 ALGORITHMS 113 beling can be used with Algorithm AЈ to give a performance ratio of max{1, (c1/2)/c0, 9c2/c0} Algorithm AЈ is based on a uniform repitition of an assignment of one channel per node, and will therefore work best when the distribution of weights in the network is fairly uniform To accommodate for nonuniform weights, Fitzpatrick et al [9] give an algorithm for hexagon graphs with parameters c0, c1, c2, where c0 = c1 and c1 Ն 2c2, which combines an assignment phase based on a labeling by arithmetic progression with two borrowing phases, in which nodes with high demand borrow unused channels from their neighbors The labeling f that is the basis of the algorithm is defined by the parameters a = c1, b = 3c1 + c2, and n = 5c1 + 3c2 It can be verified that f indeed satisfies the constraints c1 and c2 It is also the case that c2 Յ f (i, j) Յ n – c2 even for nodes (i, j) at graph distance of (0, 0) So, any channel assignment derived from f has the property that the nodes at graph distance also have separation at least c2 (This implies that the given labeling satisfies the constraints c1, c2, c3; in fact, when c1, c2 = 2, 1, the labeling is the same as the one given in [3].) More precisely, v can calculate T(v), where Ά Α w(u) | C a clique, d(u, v) Յ for all u ʦ C· T(v) = max uʦC The algorithm then proceeds in three phases, as described below Phase Node v receives channels f (v) + in, Յ i < min{w(v), T(v)/3} Phase If v has weight higher than T(v)/3, then v will borrow any unused channels from its neighbor x = (i + 1, j) Phase If v still has unfulfilled demand after the last phase, then v borrows the remaining channels from its neighbor y = (i + 1, j – 1) The algorithm can be implemented in a distributed manner Every node v = (i, j) knows its value under f, f (i, j), and is able to identify its neighbors and their position with respect to itself, and receive information about their weight Specifically, v is able to identify the neighbors (i + 1, j) and (i + 1, j – 1), and to calculate the maximum weight on a clique among its neighbors Using lower bound 5.1, applied to a 2-clique of the graph, it can be shown that the performance ratio of this algorithm equals 5/3 + c1/c2 5.3.4 Other Graphs For general graphs, a method to obtain graph labelings is to assign channels to nodes greedily The resulting span depends heavily on the order in which nodes are labeled, since each labeled node at graph distance i from a given node disqualifies 2ci – possible labels for that node Given the ordering, a greedy labeling can be found in linear time Almost all work involving greedy labelings has been done for constraints c1, c2 = 2, In this section we will assume that the constraints are these, unless otherwise noted 114 CHANNEL ASSIGNMENT AND GRAPH LABELING Any labeling for the given constraints will have span at least ⌬ + 1, as can be deduced from examining a node of maximum degree and its neighbors It can be deduced from Brooks’ theorem (see [8]), that each graph G with maximum degree ⌬ has a labeling with span at most ⌬2 + 2⌬ Griggs and Yeh [15] observe that trees have a labeling of span at most ⌬ + Nodes are labeled so that nodes closer to the root come first Each unlabeled node then has at most one labeled neighbor, and at most ⌬ – labeled nodes at distance from it The authors conjecture that it is NP-hard to decide whether a particular tree has minimum span ⌬ + or ⌬ + This conjecture was proven false by Chang and Kuo [5] Sakai [32] uses a perfect elimination ordering to show that chordal graphs have a labeling of span at most (⌬ + 3)2/4 A perfect elimination ordering v1, v2, , of the nodes has the property that for all i, Յ i Յ n, the neighbors of vi in the subgraph induced by v1, v2, , vi–1 form a clique A similar approach was later used by Bodlaender et al [4] to obtain upper bounds on labelings of graphs with fixed tree width Planar graphs are of special interest in the context of channel assignment, since a graph representing adjacency relations between cells will necessarily be planar In [38], van den Heuvel and McGuinness use methods such as used in the proof of the four color theorem to prove that all planar graphs with constraints c1, c2 admit a graph labeling of span at most (4c1 – 2)⌬ + 10c2 + 38c1 – 23 5.4 CONCLUSIONS AND OPEN PROBLEMS I have given an overview of channel assignment algorithms that take channel spacing constraints into consideration I have also reviewed the lower bounds and lower bounding techniques available for this version of the channel assignment problem Many of the algorithms described are based on graph labeling, hence an overview of relevant results on graph labeling is included in this exposition All the algorithms reviewed in this chapter have proven performance ratios Very little is known about the best possible performance ratio that can be achieved A worthwhile endeavor would be to find lower bounds on the performance ratio of any channel assignment algorithm for specific graphs and/or specific constraint parameters Other types of constraints may arise in cellular networks Many cellular systems operate under intermodulation constraints, which forbid the use of frequency gaps that are multiples of each other Channel assignment under intermodulation constraints is related to graceful labeling of graphs Another type of constraint forbids the use of certain channels in certain cells Such constraints may be external, resulting from interference with other systems, or internal, when an existing assignment must be updated to accomodate growing demand This problem is related to list coloring In practice, the most commonly encountered channel separation constraints are cosite constraints and intersite constraints of value or This situation corresponds to a constrained graph with parameters c0, c1, , ck, where c1 = = cj = and cj+1 = = ck = Much work on graph labelings focusses on constraints and 2, most specifically, constraints c1, c2 = 2, and c1, c2, c3 = 2, 1, As shown above, graph labelings can be repeated to accomodate demands of more than one channel per node It would be useful to see if REFERENCES 115 there are any better ways to use these 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used,... same node or to different nodes More precisely, ci represents the constraint between nodes at graph distance i from each other The reuse distance of G equals k + 1, the minimum graph distance between... of one, which indicates that channels assigned to the corresponding cells must be distinct, is often referred to as a cochannel constraint An intersite constraint of two, which codifies the requirement